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Chapter 2: Problem 76

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(g(x)=[f(x)]^{-2}\), where \(f\) is differentiable, then $$g^{\prime}(x)=-\frac{2 f^{\prime}(x)}{[f(x)]^{3}}$$

Short Answer

Expert verified
The statement is true, as after applying the chain rule to find the derivative of \(g(x)\), we obtained the expression $$g^{\prime}(x) = -\frac{2 f^{\prime}(x)}{[f(x)]^{3}},$$ which matches the given statement.

Step by step solution

01

Apply the chain rule

Let's find the derivative of \(g(x)\) using the chain rule. The chain rule states that if there's a composite function \(h(x) = (f \circ g)(x) = f(g(x))\), then the derivative is $$h^{\prime}(x) = f^{\prime}(g(x)) \cdot g^{\prime}(x)$$ Now, we can rewrite \(g(x) = [f(x)]^{-2}\) as \(g(x) = f(x)^{-2}\). Then, finding the derivative of \(g(x)\) using the chain rule, we have: $$g^{\prime}(x) = -2f(x)^{-3} f^{\prime}(x).$$
02

Compare the result with the given statement

Now, let's compare our result with the given statement: $$g^{\prime}(x) = -\frac{2 f^{\prime}(x)}{[f(x)]^{3}}.$$ Rearranging the result we got in step 1, we have: $$g^{\prime}(x) = -2f(x)^{-3} f^{\prime}(x) = -\frac{2 f^{\prime}(x)}{[f(x)]^{3}}.$$ Since both expressions match, we conclude that the statement is true.

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